Let $V$ be a real vector space (of finite dimension) and let $G$ be a unipotent Lie subgroup of $\mathrm{GL}(V)$. The orbits of points under the action of $G$ (that is, the sets $Gx = \{T(x) \ : \ T \in G\}$ where $x \in V$) are algebraic varieties --- this follows from Kostant-Rosenlicht theorem. I am wondering whether more is true:

**Question:** Let $x \in V$. Do there exist polynomial maps $f_1,\dots,f_s \colon V \to \mathbb{R}$ whose zero locus is $Gx$ and which are invariant under the action of $G$?
(i.e.: $Gx = \{ y \in V \ : \ f_i(y) = 0 \text{ for } i = 1,2,\dots,s\}$ and $f_i(Ty) = f(y)$ for all $T \in G,\ y \in V$.)